Here's how I'd break it down:
Flat Tarp: Just a big, flat rectangle of fabric. Pretty easy to make, pretty cheap to buy. Can be pitched in all sorts of origami-like configurations. Some people like learning a bunch of different pitches, and like the concept that a simple tarp can be used in so many different ways. In principle, so do I. In practice, not so much.
Catenary-cut Tarp: Has a catenary curve sewn into the long ridge line, which makes getting a taut pitch much easier. Pretty much limited to a-frame configuration. Without doors or beaks, can the lightest option. Designing the cat-cut is a little tricky, but otherwise about as easy to make as a flat tarp. Plenty of ready-made versions, too.
Shaped Tarp: Think tent fly or single-wall tent, but without an integrated floor (use a ground sheet or bivy). Typically can be pitched low to the ground for 360* protection, meaning no splash-resistent bivy required. Usually only one basic way to pitch, but should be just as easy to pitch as a well-designed tent. Often three or more panels of fabric, multiple cat-cut seams, and zippered door(s), although can be pretty simple as well (see pic below). Typically harder to make, more expensive to purchase. Plenty of designs out there: SMD Wild Oasis, MLD 'Mids and Trailstar, to name only a few. Here's one I made (two pieces of 1.1 silnylon fabric, no door, 9.7 oz weight including guylines):